I doubt this is a suitable problem for a novice. even showing convergence is tough. i will look at the other posted answers. there is a good reason people start from the integral definition of ln(x) to derive this result.

The derivative is lim (f(x+h)- f(x))/h= axlim {(ah-1)/h}. Notice that that is ax time a limit that is independent of x. That is, as long as the derivative exists, it is ax times a constant. The problem is showing that the lim{(ah-1)/h} EXISTS! And then showing that, if a= 2, that limit is ln(2).

Showing that that limit exists is sufficiently non-trivial that many people (myself included), as mathwonk said, prefer to define ln(x) as the integral, from 1 to x of (1/t)dt. From that, it is possible to prove all properties of ln(x) including (trivially) that the derivative is 1/x. Defining ex as the inverse function of ln(x) leads to all the properties of ex (including the fact that it is some number to a power!), in particular that its derivative is ex itself and, from that, that the derivative of ax is (ln a) ax.

No, you did exactly what HallsofIvy was advocating, he was just pointing out that the question asked for it to be solved using the definition of a derivative, which makes things much harder. Easier to approach things from the other way, starting by defining the integral of 1/x.

"The derivative is lim (f(x+h)- f(x))/h= axlim {(ah-1)/h}. Notice that that is ax time a limit that is independent of x. That is, as long as the derivative exists, it is ax times a constant. The problem is showing that the lim{(ah-1)/h} EXISTS! And then showing that, if a= 2, that limit is ln(2)."

"The derivative is lim (f(x+h)- f(x))/h= axlim {(ah-1)/h}. Notice that that is ax time a limit that is independent of x. That is, as long as the derivative exists, it is ax times a constant. The problem is showing that the lim{(ah-1)/h} EXISTS! And then showing that, if a= 2, that limit is ln(2)."